Large time decay properties of solutions to a viscous Boussinesq system in a half space

Abstract

We consider the long time behavior of weak and strong solutions of the n-dimensional viscous Boussinesq system in the half space, with $n\geq3$ . The $L^r(R^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives the unboundedness of the projection operator $P: L^\infty(R^n_+)\rightarrow L^\infty_\sigma(R^n_+)$ is dealt by an appropriate decomposition of the nonlinear term.